At age 20 in Dublin, Oliver Byrne made his author’s debut with *A Treatise on Diophantine Algebra.* Although most often referenced as *Treatise on Algebra,* in his 1858 Royal Literary Fund (RLF) application, Byrne listed this volume specifically as *A Treatise on Diophantine Algebra,* published by Allen and Co., Dublin, 1830. Second, he published *A Pamphlet on the Teaching of Geometry by Coloured Diagrams, etc.; Applied to the First Book of Euclid* in 1831*.*^{10} By 1838, Byrne had expanded this pamphlet into a treatment of the first six books of Euclid's *Elements,* advertised in another of his publications as

"*In the Press,*

The second Edition of The Elements of Geometry, containing the first six Books of Euclid, in which the propositions are demonstrated by the substitution of *colours* for *letters,* the study being thus shortened and improved.

*Price Eight Shillings.*"

This book apparently never reached a seller; however, Byrne's famous *The First Six Books of the Elements of Euclid in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners* was finally published in 1847 in London by William Pickering. An in depth discussion of Byrne’s lasting achievement, his *Euclid,* follows in a later section of this article, Byrne's *Euclid:* Geometry Understood via Color-coded Diagrams.

**Figure 2.** Title page of Oliver Byrne's *The First Six Books of the Elements of Euclid in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners, *published in 1847. This book grew out of a pamphlet by Byrne containing just Book I of Euclid's *Elements* presented using colorful diagrams, published in 1831 when he was just 21 years old. (The image above is from the book in the collection of Sid Kolpas.)

Byrne’s other early publications included *How to Measure the Earth with the Assistance of Railroads* and *New and Improved System of Logarithms,* both in 1838, as well as a variety of mathematical pamphlets. A comprehensive list of Byrne’s known works is given in Appendix A of this article.

Not only a mathematician and author, Byrne also invented scientific instruments. On 10 March 1838, Abraham Parker, Surveyor, and Oliver Byrne, Professor of Mathematics, both of Gower Street, Bedford Square, London, filed for a patent covering an “Instrument for gauging malt and the fluid or solid content of casks and other vessels.”^{11} In 1843 Byrne wrote a “Description and use of an instrument to find the time by the sun, moon, or any of the visible fixed stars: as well as the names of those stars / invented and constructed for the Right Hon. Earl Fitzwilliam.”^{12} Later, in 1846, Byrne wrote of another invention, the Byrnegraph. The Byrnegraph was, according to Oliver Byrne, an improved proportional compass, designed on precise mathematical principles to remedy “the defects of the old proportional compasses” and extend

the application of the instrument of so much importance to the engineer, the artist, the architect, and the amateur, in every department of applied mathematics, when its results can be relied on with confidence.

Byrne claimed that traditional proportional compasses scarcely deserved the title “mathematical,” but that his version was constructed on purely mathematical principles, thus providing much greater accuracy.^{13}

Using the pseudonym E. B. Revilo (Oliver spelled backward), Byrne attempted to prove the truth of the Creed of Saint Athanasius through a mathematical analogy in a pamphlet titled *The Creed of Saint Athanasius proved by a Mathematical Parallel *and published in 1839.^{14} According to historian Daniel Cohen, Byrne sought to equate^{15}

each element of the Trinity (the Father, the Son, and the Holy Ghost) to infinity, since the Athanasian Creed considered all of them to be unlimited. Byrne then erected two vertical columns: the left containing the English Book of Common Prayer translation of the Quicunque Vult (the traditional description of the Athanasian Creed), the right containing parallel mathematical equations involving infinity that purported to establish the truth of the statements on the left.

The mathematician Augustus De Morgan (1806-1871), in *A Budget of Paradoxes,* judged this work to be nonsense and considered Byrne to be an eccentric mathematician.^{16} Janet Heine Barnett provided further analysis of DeMorgan’s writing on the works of Oliver Byrne in a recent article.^{17} Byrne also reviewed the work of his colleagues.^{18}

**Figure 3.** Title page of Augustus De Morgan's *A Budget of Paradoxes* (1872), a collection of the author's amusing and often critical writings about the attempted mathematical works of others. (This image is presented courtesy of the University of Pennsylvania Libraries and may be used in your classroom; for all other purposes, please seek permission from the University of Pennsylvania Libraries.)

Financial and legal problems plagued Oliver Byrne periodically throughout his entire life. His first of fifteen financial aid applications to the Royal Literary Fund occurred in 1839 while jailed as a debtor at White Cross Street Prison, London. On 23 November 1839, Byrne wrote,^{19}

I have been a resident in London for the last 8 years, during which time I have supported myself as a teacher of Mathematics, and have also published several works in connection with the mathematical sciences …. A native of the sister country I have had to struggle long and severly [sic] with adverse circumstances, and to encounter an almost overwhelming weight of prejudice, but my pupils have included some of the most distinguished in the ranks of society …. I have now unfortunately been arrested on account of a debt of thirty pounds which I have not been able to pay and now lay here at the suit of a merciless creditor whom I have not the means to satisfy.

Perhaps an example of Byrne’s claim of prejudice against the Irish occurred when a friend came to bail Byrne out in 1839 but it was refused by the plaintiff, even after the friend offered an even higher amount as security. The London newspaper *The Examiner *reported,^{20}

A person of the name of Oliver Byrne, a professor of mathematics, appeared in Court to be discharged on bail under the New Insolvent Debtors’ Act of last sessions.—His bail appeared in Court to justify, but one of them was opposed by Mr Woodruff, on behalf of the plaintiff, as to the bail’s sufficiency. The person opposed was pressed hard by counsel as to his means, and at last refused to answer any more questions, his feeling being much irritated at the interrogatories. The other bail, finding his co-bail in danger of passing as one of the sureties, then offered to pay the required sum into court as security for the insolvent’s appearance on the final day of hearing, rather than he should be inconvenienced by staying in prison for about five weeks.—This offer was refused by the Court. [This appears to us extremely unreasonable, vexatious, and unjust. The payment into Court of the sum required in security was obviously as good as or better than the best bail, and we are at a loss to conceive the pretext on which the offer can have been refused.]

Beginning with his first grant in 1839, the Royal Literary Fund’s support of Oliver Byrne amounted to £320 between 1839 and 1881 when his wife applied as his widow.^{21} David Williams founded The Royal Literary Fund in 1790 to “withdraw those apprehensions of extreme poverty, and those desponding views of futurity, which lead Genius and Talent from the path of Virtue.” Other Royal Literary Fund beneficiaries included novelists Bram Stoker, Charles Dickens, Jr., and James Joyce.^{22}

For a short time, Byrne taught mathematics in the College of Civil Engineers at Putney in South West London.^{23} Founded for the purpose of affording sound instruction in the theory and practical application of civil engineering and architecture, the College of Civil Engineers prepared students for a career in the field.^{24} Started in 1838 in Kentish Town, the school moved to Putney in August 1840.^{25} Putney House, one of two large mansions in the area, was converted into a College for Civil Engineers in 1839, founded by subscriptions among the nobility and others “for the purpose of conferring a superior education on the sons of respectable persons in the engineering, mathematical, and mechanical sciences.” The college closed in 1857 and the mansion was demolished.^{26}

Byrne adopted the specific title of “Professor of Mathematics at the College for Civil Engineers” (versus simply “Professor of Mathematics”) in August 1839.^{27} This was also the year of his first Royal Literary Fund (RLF) application, submitted from debtor’s prison due to a debt owed to an unspecified creditor. His brother, John Byrne, reported in *Miscellaneous Mathematical Papers of Oliver Byrne, Collected and edited by John Byrne,* that “lectures [were] delivered by Professor Oliver Byrne, at the Museum Lecture room, Philadelphia, in January 1842, . . ..”^{28} Thus, it appears that Byrne’s tenure at the College for Civil Engineers may have lasted just over two years, from August 1839 to December 1841. Concerning his resignation from his post at Putney College, Byrne himself later explained,^{29}

I resigned the professorship of Mathematics in the College for Civil Engineers, not without sufficient reason; the most persevering, in my position, would have done the same. Several of the Council resigned for the same reason. After my resignation, I went to the United States of America, hoping that I could stipulate with the government for a new plan of calculating the revenue. I travelled from City to City without success supporting myself by delivering lectures on popular mathematical subjects and by writing short articles for periodicals and newspapers.

10 Byrne assigned Dublin and 1832 and 1833 to these publications in their earliest known reference, Royal Literary Fund (RLF) application (28 May 1842). In RLF application (6 January 1858), Byrne entitled his first work *A Treatise on Diophantine Algebra* published by Allen & Company, Dublin, 1830, and he noted it “__very scarce__.” There is evidence of a Quaker printer, Richard Allen, active in Dublin in this time; see Bruce Nelson, *Irish Nationalists and the Making of the Irish Race* (Princeton: Princeton University Press, 2012), 99. The notice of Byrne's "second Edition of The Elements of Geometry, containing the first six Books of Euclid ..." appeared at the end of his 1838 publication, *How to Measure the Earth with the Assistance of Railroads**,* available at GoogleBooks.

11 David J. Bryden (Worcestershire, England) to Susan Hawes (Portland, Maine), "Oliver Byrne," email, 17 December 2014, Byrne Research Files; privately held by Susan Hawes, Portland, Maine. David Bryden spent most of his career as a museum curator. After his retirement, Bryden continues to research early scientific instruments. (“London meeting lecture Thursday 15 November 2012 Bankruptcy and insolvency in the English horological trade, 1720-1849 by David Bryden," newsletter; Antiquarian Horological Society (http://www.ahsoc.org : accessed 21 June 2015).

12 Oliver Byrne, "Description and use of an instrument to find the time by the sun, moon, or any of the visible fixed stars: as well as the names of those stars / invented and constructed for the Right Hon. Earl Fitzwilliam," manuscript, 1843; Harvard University Houghton Library.

13 Oliver Byrne, *Description and Use of the Byrnegraph: an instrument for multiplying, dividing, and comparing lines, angles, surfaces, and solids* (London: C. & J. Adlard, Bartholomew Close, 1846), 3.

14 Oliver Byrne, *The Creed of Saint Athanasius proved by a Mathematical Parallel* (London: William Day, 1839).

15 Daniel J. Cohen, *Equations from God: Pure Mathematics and Victorian Faith* (Baltimore: John Hopkins University Press, 2007), 73.

16 Augustus DeMorgan, *A Budget of Paradoxes* (London: Longmans Green & Co, 1872), 329.

17 Janet Heine Barnett, “Mathematics is a Plural Noun: The Case of Oliver Byrne,” *Proceedings of the Canadian Society for the History and Philosophy of Mathematics: Thirty-fifth Annual Meeting (Montreal),* 23:26-46 (2010).

18 Oliver Byrne, “Unwillingness of Man to Investigate,” *American Railroad Journal and Mechanics Magazine,* Vol VIII—New Series or Vol. XIV, 91-93 (New York: George C. Schaeffer, 1842).

19 RLF application (23 November 1839).

20 "Insolvent Debtors' Court, Monday," *The Examiner* (London), 29 December 1839.

21 MeasuringWorth.com: In 2013, the relative value of £320 in 1875 ranges from £26,300.00 to £407,100.00. Using Google to calculate British pounds into 2015 US dollars, this amount reflects a value of $41,000 - $639,000. Eleanor Byrne, widow of Oliver Byrne, RLF application (3 January 1881).

22 Archives of the Royal Literary Fund; Nineteenth Century Collections Online: "British Politics and Society” Collection; Gale Digital Collections (http://www.galegroup.com : 8 August 2014). For other beneficiaries, see “The RLF Archive,” Royal Literary Fund, (http://www.rlf.org.uk/home/the-rlf-archive : accessed 21 December 2014).

23 RLF application (23 November 1839). “1841 England Census,” [county] Surrey, [parish] Putney, enumeration district 5, folio 62, line 10, Oliver Byrne, professor; *Ancestry* (http://www.ancestry.com : accessed 30 July 2014).

24 Edward Walford, *Old and New London: A Narrative of Its History, Its People and Its Places. The Southern Suburbs Vol VI* (http://www.british-history.ac.uk : accessed 7 September 2014).

25 John Timbs, *Curiosities of London: Exhibiting the Most Rare and Remarkable Objects of Interest in the Metropolis* (London: D. Bogue, 1855).

26 Edward Walford, *Old and New London: A Narrative of Its History, Its People and Its Places. The Southern Suburbs Vol VI* (http://www.british-history.ac.uk : accessed 7 September 2014).

27 “British Association Fracas,” *The Mechanic's Magazine,* *Museum, Register, Journal, and Gazette,* April 6th - September 28th, 1939, Vol. XXXI (London: W.A. Robertson, 1839).

28 John Byrne, *The Miscellaneous Mathematical Papers of Oliver Byrne, Collected and edited by John Byrne* (London: Maynard, 1848), cited in Janet Heine Barnett, “The Dual Arithmetic of Oliver Byrne: A New Art which entirely supersedes the use of logarithms,” *Proceedings of the Canadian Society for the History and Philosophy of Mathematics: Thirty-sixth Annual Meeting (Dublin),* 24:12-30 (2011).

29 RLF application (28 May 1842). Example article Byrne wrote during this period, Oliver Byrne, “Unwillingness of Man to Investigate,” *American Railroad Journal and Mechanics Magazine,* Vol. VIII—New Series or Vol. XIV, 91-93 (New York: George C. Schaeffer, 1842).